Relaxation versus Decoherence: Spin and Current Dynamics in the Anisotropic Kondo Model at Finite Bias and Magnetic Field

Using a nonequilibrium renormalization group method we study the real-time evolution of spin and current in the anisotropic Kondo model (both antiferromagnetic and ferromagnetic) at a finite magnetic field $h_0$ and bias voltage $V$. We derive analytic expressions for all times in the weak-coupling regime $\max \{V,h_0,1/t\}\gg T_c$ ($T_c$ is the strong-coupling scale). We find that all observables decay both with the spin relaxation and decoherence rates ${\Gamma }_{1/2}$. Various $V$-dependent logarithmic, oscillatory, and power-law contributions are predicted. The low-energy cutoff of logarithmic terms is generically identified by the difference of transport decay rates. For small times $t\ll \max \{V,h_0{\}}^{-1}$, we obtain universal dynamics for spin and current.

Figure 1

A spin-$\frac{1}{2}$ quantum system coupled via exchange couplings $J_{z/\perp }$ to the spins of two reservoirs.

Figure 2

$S_z(t)$ in the isotropic Kondo model for $h_0=100T_K$ and various values of the applied voltage $V$, with $S_z(0)=1/2$. Inset: $S_x(t)$ for $V=h_0=100T_K$ and $S_x(0)=1/2$, $S_y(0)=0$.

Figure 3

Current $I(t)$ in the isotropic Kondo model for $h_0=100T_K$ (solid lines) and $h_0=200T_K$ (dashed lines) and various values of the applied voltage $V$, with $S_z(0)=1/2$. We observe oscillations at times $T_{K}t\sim 0.05$; the stationary current is reached for $T_{K}t\ge 0.5$.

Figure 4

${log}_{10}\mathbf{(}|S_x(t)|\mathbf{)}$ in the anisotropic Kondo model for $V=2h_0=100T_K$ and various values of the anisotropy $c^2=(J_z^{(0)}{)}^2-(J_{\perp }^{(0)}{)}^2$, with $S_x(0)=1/2$, $S_y(0)=0$. The dips have their origin in the oscillations of $S_x(t)$.